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### Course: Algebra 2>Unit 8

Lesson 4: The change of base formula for logarithms

# Using the logarithm change of base rule

Sal rewrites logarithmic expressions like 1/(logₐ4) or logₐ(16)*log₂(a) using the change of base rule.

## Want to join the conversation?

• How can I take this one step further to solve, say, 4^x=9?

Got it. If anyone else is curious, here’s my answer:
Okay, so I’ve got that this equals log_4 (9)=x. Then, I just have to use the change of base formula to change it to log_10 so that I can enter it in my calculator.

log_10 (9)
---------------
log_10 (4)
• Yes, I think you're correct. You get log_4(9) for an answer.
• What does logarithms even mean? For example log(15) is about 1.176. What does that mean? How do I use it?
• Usually, log means there's a hidden base of 10 (though sometimes in much higher levels of math, log means a hidden base of e). I will assume a base of 10 here.

Since the answer to a logarithm problem is an exponent, log(15) is about 1.176 means that 10^1.176 is about 15. I have confirmed this on a calculator.

Logarithms have several uses in the real world, such as the pH scale for acidity in chemistry, the Richter scale for magnitudes of earthquakes, the decibel scale for loudness of sounds, and determining how long the balance in a compound interest savings account would take to reach a certain value.

Have a blessed, wonderful day!
• What if the denominator has no base (e.g. log(6))? what then?
• log(6) usually means log_10(6), the 10 is usually omitted, though if you mean literally having no base, then its not possible. Since logarithm is basically finding to which power a number(base) must be raised to get another number. If there is no base, then there is no log of that base either
• what is the difference between log(m/n) and log m/log n?
• log(m/n) is the quotient or difference rule
log(m/n) = log(m) - log(n)

log(m) / log(n) is change of base rule.
log(m) / log(n) = log_n(m) . Read log base n of m.
• So the base of (logb)/(log4) can be anything?
• yes, you can use any base and log(a)/log(b) will still equal log_b(a).
• Shouldn't the 1/log_a b = log_b a be a logarithm property? If you know it as a rule it would save you quite a bunch of time when solving problems
• I have a question, say in a test your teacher ask you to simply 1/logb(4) using the change of base rule
and you did that, but not using sal's approach, you didn't go with 1/(log4/logb). but going with 1/(log2[4]/log2[b]). which still change of base rule.
and you get log2(b)/2, or even further you may get log2(b^2) as the answer
now my question is, (assuming I didn't make any arithmetic mistake above) will this be a "correct" answer?
consider what they asked is "using the change of base rule to simplify"
• I'm not sure how you made that last step. `log2(b) / 2 = log2(b) / log2(4) = log4(b)` . To get `log2(b^2)` you'll have to arbitrarily multiply `log2(b) / 2` by 4, which is not a legal operation.

Regardless, you can use any base you want, unless some particular base allows to simplify the expression further. For example, if you have an expression like `logb(c) * log(b)` changing base of `logb(c)` to 10 allows you to simplify the expression down to `log(c)`, while using some other base does not.

Also, you might be asked to give an exact answer using a calculator, and most likely it will only have `log` and `ln` functions, so using some other base wouldn't make much sense.
• At , how can Sal simply cancel out the log_10 (C)?
As log_10 (C) is in the denominator, does it matter whether log_10 (C) will equal zero or not?
Thank you :)
• First, he canceled the out because log(C)/log(C)=1, and anything times 1 is itself. Second, log(x), if x is anything, can't equal 0 because nothing with an exponent equals 0, unless 0 is the base, but, by default, log's base is 10
(1 vote)
• when do I round up to the thousand? do I round up when I solve the operation on the numerator and also on the denominator or do I keep the full numbers and round up only the final answer? I got several operations wrong because of this
• You round up to the thousandth at the very end of the operation. Do not round up anything whilst you are doing the expression!

It is much more better than doing the expression in one, full long line in a calculator rather than separated segments. Or you may also use the "ans" key that preserves your answer in your new calculator line.